Feuilletons & Causeries on a Variety of Subjects

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I have just finished reading Amir D. Aczel’s book on the mathematics of infinity, entitled The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity (New York: Washington Square Press, 2000). Although I am an ignoramus when it comes to theoretical mathematics, I was able to follow this book pretty much throughout; and I found it utterly fascinating.

The most important contributions to the mathematics of infinity have come from two Europeans, first Georg Cantor of Germany and then Kurt Gödel of Austria. In vainly attempting to prove their theorems regarding infinite sets, both men went mad. Cantor died at Halle’s Nervenklinik in 1918; and Gödel of starvation in Princeton, NJ, of all places, in 1978.

Georg Cantor

What sent Cantor to the clinic multiple times were his difficulties in finding a proof for the so-called Continuum Hypothesis, which, stated all too briefly, is that there is no set whose cardinality is strictly between that of the integers and the real numbers. In other words, it relates to quantifying systems of sets containing infinite values, whether real or integer. Got that? Well, it killed Cantor and also Gödel.

Kurt Gödel

Even though at the end his mind was wasted, and he insisted that people were trying to poison him, Gödel finally understood one very important fact:

Gödel and [Paul] Cohen have brought us to a sobering realization: hard as we may try, there will always be some truths forever beyond our reach. Human beings may never understand the deep nature of infinity. This is perhaps something that Kabbalah practitioners understand on an intuitive level, without requiring a mathematical proof. To them, infinity was God or things that are God’s. One such infinity was the chaluk, God’s infinitely bright robe, at which no human can look.

It all makes sense. To prove something, one has to be able to see through to its essence, which is difficult when infinities are involved.